The flow in a channel with an oscillating constriction has been studied by the numerical solution of the Navier-Stokes and Euler equations. A vorticity wave is found downstream of the constriction in both viscous and inviscid flow, whether the downstream flow rate is held constant and the upstream flow is pulsatile, or vice versa. Closed eddies are predicted to form between the crests/troughs of the wave and the walls, in the Euler solutions as well as the Navier-Stokes flows, although their structures are different in the two cases.
The positions of wave crests and troughs, as determined numerically, are compared with the predictions of a small-amplitude inviscid theory (Pedley & Stephanoff 1985). The theory agrees reasonably with the Euler equation predictions at small amplitude (ε [lsim ] 0.2) as long as the downstream flow rate is held fixed; otherwise a sinusoidal displacement is superimposed on the computed crest positions. At larger amplitude (ε = 0.38) the wave crests move downstream more rapidly than predicted by the theory, because of the rapid growth of the first eddy (‘eddy A’) attached to the downstream end of the constriction. At such larger amplitudes the Navier-Stokes predictions also agree well with the Euler predictions, when the downstream flow rate is held fixed, because the wave generation process is essentially inviscid and the undisturbed vorticity distribution is the same in each case. It is quite different, however, when the upstream flow rate is fixed, as in the experiments of Pedley & Stephanoff, because of differences in the undisturbed vorticity distribution, in the growth rate of the vorticity waves and in the dynamics of eddy A. A further finite-amplitude effect of importance, especially in an inviscid fluid, is the interaction of an eddy with its images in the channel walls.